\(\int \frac {(A+B x) \sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1830]

Optimal result

Integrand size = 33, antiderivative size = 313 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-7 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}} \]

[Out]

-1/5*(A*b-B*a)*(e*x+d)^(3/2)/b/(-a*e+b*d)/(b*x+a)^5+1/128*e^4*(-7*A*b*e-3*B*a*e+10*B*b*d)*arctanh(b^(1/2)*(e*x
+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(5/2)/(-a*e+b*d)^(9/2)-1/40*(-7*A*b*e-3*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^2/(-a*e+
b*d)/(b*x+a)^4-1/240*e*(-7*A*b*e-3*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^2/(-a*e+b*d)^2/(b*x+a)^3+1/192*e^2*(-7*A*b*
e-3*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^2/(-a*e+b*d)^3/(b*x+a)^2-1/128*e^3*(-7*A*b*e-3*B*a*e+10*B*b*d)*(e*x+d)^(1/
2)/b^2/(-a*e+b*d)^4/(b*x+a)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 79, 43, 44, 65, 214} \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^4 (-3 a B e-7 A b e+10 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac {e^3 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac {e^2 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}-\frac {e \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac {\sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[In]

Int[((A + B*x)*Sqrt[d + e*x])/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/40*((10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(b^2*(b*d - a*e)*(a + b*x)^4) - (e*(10*b*B*d - 7*A*b*e -
3*a*B*e)*Sqrt[d + e*x])/(240*b^2*(b*d - a*e)^2*(a + b*x)^3) + (e^2*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x
])/(192*b^2*(b*d - a*e)^3*(a + b*x)^2) - (e^3*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(128*b^2*(b*d - a*
e)^4*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(3/2))/(5*b*(b*d - a*e)*(a + b*x)^5) + (e^4*(10*b*B*d - 7*A*b*e - 3*a
*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(5/2)*(b*d - a*e)^(9/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^6} \, dx \\ & = -\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-7 A b e-3 a B e) \int \frac {\sqrt {d+e x}}{(a+b x)^5} \, dx}{10 b (b d-a e)} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(e (10 b B d-7 A b e-3 a B e)) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 b^2 (b d-a e)} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^2 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{96 b^2 (b d-a e)^2} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^3 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^2 (b d-a e)^3} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^4 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^2 (b d-a e)^4} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^3 (10 b B d-7 A b e-3 a B e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^2 (b d-a e)^4} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-7 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.46 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {d+e x} \left (B \left (-45 a^5 e^4+30 a^4 b e^3 (4 d-7 e x)+2 a^3 b^2 e^2 \left (-218 d^2+409 d e x+192 e^2 x^2\right )-10 b^5 d x \left (48 d^3+8 d^2 e x-10 d e^2 x^2+15 e^3 x^3\right )+2 a^2 b^3 e \left (176 d^3-1178 d^2 e x-709 d e^2 x^2+105 e^3 x^3\right )+a b^4 \left (-96 d^4+1808 d^3 e x+484 d^2 e^2 x^2-730 d e^3 x^3+45 e^4 x^4\right )\right )+A b \left (-105 a^4 e^4+10 a^3 b e^3 (121 d+79 e x)+2 a^2 b^2 e^2 \left (-1052 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (744 d^3+128 d^2 e x-161 d e^2 x^2+245 e^3 x^3\right )+b^4 \left (-384 d^4-48 d^3 e x+56 d^2 e^2 x^2-70 d e^3 x^3+105 e^4 x^4\right )\right )\right )}{(b d-a e)^4 (a+b x)^5}+\frac {15 e^4 (-10 b B d+7 A b e+3 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}}}{1920 b^{5/2}} \]

[In]

Integrate[((A + B*x)*Sqrt[d + e*x])/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((Sqrt[b]*Sqrt[d + e*x]*(B*(-45*a^5*e^4 + 30*a^4*b*e^3*(4*d - 7*e*x) + 2*a^3*b^2*e^2*(-218*d^2 + 409*d*e*x + 1
92*e^2*x^2) - 10*b^5*d*x*(48*d^3 + 8*d^2*e*x - 10*d*e^2*x^2 + 15*e^3*x^3) + 2*a^2*b^3*e*(176*d^3 - 1178*d^2*e*
x - 709*d*e^2*x^2 + 105*e^3*x^3) + a*b^4*(-96*d^4 + 1808*d^3*e*x + 484*d^2*e^2*x^2 - 730*d*e^3*x^3 + 45*e^4*x^
4)) + A*b*(-105*a^4*e^4 + 10*a^3*b*e^3*(121*d + 79*e*x) + 2*a^2*b^2*e^2*(-1052*d^2 - 289*d*e*x + 448*e^2*x^2)
+ 2*a*b^3*e*(744*d^3 + 128*d^2*e*x - 161*d*e^2*x^2 + 245*e^3*x^3) + b^4*(-384*d^4 - 48*d^3*e*x + 56*d^2*e^2*x^
2 - 70*d*e^3*x^3 + 105*e^4*x^4))))/((b*d - a*e)^4*(a + b*x)^5) + (15*e^4*(-10*b*B*d + 7*A*b*e + 3*a*B*e)*ArcTa
n[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(9/2))/(1920*b^(5/2))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(-\frac {7 \left (-\left (\left (A e -\frac {10 B d}{7}\right ) b +\frac {3 B a e}{7}\right ) \left (b x +a \right )^{5} e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+\sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, \left (\left (-A \,e^{4} x^{4}+\frac {2 \left (\frac {15 B x}{7}+A \right ) x^{3} d \,e^{3}}{3}-\frac {8 x^{2} d^{2} \left (\frac {25 B x}{14}+A \right ) e^{2}}{15}+\frac {16 x \left (\frac {5 B x}{3}+A \right ) d^{3} e}{35}+\frac {128 d^{4} \left (\frac {5 B x}{4}+A \right )}{35}\right ) b^{5}-\frac {496 \left (\frac {245 \left (\frac {9 B x}{98}+A \right ) x^{3} e^{4}}{744}-\frac {161 \left (\frac {365 B x}{161}+A \right ) x^{2} d \,e^{3}}{744}+\frac {16 x \,d^{2} \left (\frac {121 B x}{64}+A \right ) e^{2}}{93}+d^{3} \left (\frac {113 B x}{93}+A \right ) e -\frac {2 B \,d^{4}}{31}\right ) a \,b^{4}}{35}+\frac {2104 \left (-\frac {112 \left (\frac {15 B x}{64}+A \right ) x^{2} e^{3}}{263}+\frac {289 x d \left (\frac {709 B x}{289}+A \right ) e^{2}}{1052}+d^{2} \left (\frac {589 B x}{526}+A \right ) e -\frac {44 B \,d^{3}}{263}\right ) e \,a^{2} b^{3}}{105}-\frac {242 \left (\frac {79 \left (\frac {192 B x}{395}+A \right ) x \,e^{2}}{121}+d \left (\frac {409 B x}{605}+A \right ) e -\frac {218 B \,d^{2}}{605}\right ) e^{2} a^{3} b^{2}}{21}+\left (\left (2 B x +A \right ) e -\frac {8 B d}{7}\right ) e^{3} a^{4} b +\frac {3 B \,a^{5} e^{4}}{7}\right )\right )}{128 \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} b^{2} \left (a e -b d \right )^{4}}\)	\(368\)
derivativedivides	\(2 e^{4} \left (\frac {\frac {\left (7 A b e +3 B a e -10 B b d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{256 e^{4} a^{4}-1024 b d \,e^{3} a^{3}+1536 b^{2} d^{2} e^{2} a^{2}-1024 b^{3} d^{3} e a +256 b^{4} d^{4}}+\frac {7 \left (7 A b e +3 B a e -10 B b d \right ) b \left (e x +d \right )^{\frac {7}{2}}}{384 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 e^{2} a^{2}-60 a b d e +30 b^{2} d^{2}}+\frac {\left (79 A b e -21 B a e -58 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b \left (a e -b d \right )}-\frac {\left (7 A b e +3 B a e -10 B b d \right ) \sqrt {e x +d}}{256 b^{2}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{2} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\)	\(389\)
default	\(2 e^{4} \left (\frac {\frac {\left (7 A b e +3 B a e -10 B b d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{256 e^{4} a^{4}-1024 b d \,e^{3} a^{3}+1536 b^{2} d^{2} e^{2} a^{2}-1024 b^{3} d^{3} e a +256 b^{4} d^{4}}+\frac {7 \left (7 A b e +3 B a e -10 B b d \right ) b \left (e x +d \right )^{\frac {7}{2}}}{384 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 e^{2} a^{2}-60 a b d e +30 b^{2} d^{2}}+\frac {\left (79 A b e -21 B a e -58 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b \left (a e -b d \right )}-\frac {\left (7 A b e +3 B a e -10 B b d \right ) \sqrt {e x +d}}{256 b^{2}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{2} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\)	\(389\)

[In]

int((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

-7/128/((a*e-b*d)*b)^(1/2)*(-((A*e-10/7*B*d)*b+3/7*B*a*e)*(b*x+a)^5*e^4*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(
1/2))+((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*((-A*e^4*x^4+2/3*(15/7*B*x+A)*x^3*d*e^3-8/15*x^2*d^2*(25/14*B*x+A)*e^2
+16/35*x*(5/3*B*x+A)*d^3*e+128/35*d^4*(5/4*B*x+A))*b^5-496/35*(245/744*(9/98*B*x+A)*x^3*e^4-161/744*(365/161*B
*x+A)*x^2*d*e^3+16/93*x*d^2*(121/64*B*x+A)*e^2+d^3*(113/93*B*x+A)*e-2/31*B*d^4)*a*b^4+2104/105*(-112/263*(15/6
4*B*x+A)*x^2*e^3+289/1052*x*d*(709/289*B*x+A)*e^2+d^2*(589/526*B*x+A)*e-44/263*B*d^3)*e*a^2*b^3-242/21*(79/121
*(192/395*B*x+A)*x*e^2+d*(409/605*B*x+A)*e-218/605*B*d^2)*e^2*a^3*b^2+((2*B*x+A)*e-8/7*B*d)*e^3*a^4*b+3/7*B*a^
5*e^4))/(b*x+a)^5/b^2/(a*e-b*d)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (281) = 562\).

Time = 0.44 (sec) , antiderivative size = 2580, normalized size of antiderivative = 8.24 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(15*(10*B*a^5*b*d*e^4 - (3*B*a^6 + 7*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (3*B*a*b^5 + 7*A*b^6)*e^5)*x^5
+ 5*(10*B*a*b^5*d*e^4 - (3*B*a^2*b^4 + 7*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (3*B*a^3*b^3 + 7*A*a^2*b
^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (3*B*a^4*b^2 + 7*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (3*B*a^
5*b + 7*A*a^4*b^2)*e^5)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))
/(b*x + a)) + 2*(96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(28*B*a^2*b^5 + 117*A*a*b^6)*d^4*e + 4*(197*B*a^3*b^4 + 898*A
*a^2*b^5)*d^3*e^2 - 2*(278*B*a^4*b^3 + 1657*A*a^3*b^4)*d^2*e^3 + 5*(33*B*a^5*b^2 + 263*A*a^4*b^3)*d*e^4 - 15*(
3*B*a^6*b + 7*A*a^5*b^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (13*B*a*b^6 + 7*A*b^7)*d*e^4 + (3*B*a^2*b^5 + 7*A*a*b^6)
*e^5)*x^4 - 10*(10*B*b^7*d^3*e^2 - (83*B*a*b^6 + 7*A*b^7)*d^2*e^3 + 2*(47*B*a^2*b^5 + 28*A*a*b^6)*d*e^4 - 7*(3
*B*a^3*b^4 + 7*A*a^2*b^5)*e^5)*x^3 + 2*(40*B*b^7*d^4*e - 2*(141*B*a*b^6 + 14*A*b^7)*d^3*e^2 + 3*(317*B*a^2*b^5
+ 63*A*a*b^6)*d^2*e^3 - (901*B*a^3*b^4 + 609*A*a^2*b^5)*d*e^4 + 64*(3*B*a^4*b^3 + 7*A*a^3*b^4)*e^5)*x^2 + 2*(
240*B*b^7*d^5 - 8*(143*B*a*b^6 - 3*A*b^7)*d^4*e + 2*(1041*B*a^2*b^5 - 76*A*a*b^6)*d^3*e^2 - 3*(529*B*a^3*b^4 -
139*A*a^2*b^5)*d^2*e^3 + 2*(257*B*a^4*b^3 - 342*A*a^3*b^4)*d*e^4 - 5*(21*B*a^5*b^2 - 79*A*a^4*b^3)*e^5)*x)*sq
rt(e*x + d))/(a^5*b^8*d^5 - 5*a^6*b^7*d^4*e + 10*a^7*b^6*d^3*e^2 - 10*a^8*b^5*d^2*e^3 + 5*a^9*b^4*d*e^4 - a^10
*b^3*e^5 + (b^13*d^5 - 5*a*b^12*d^4*e + 10*a^2*b^11*d^3*e^2 - 10*a^3*b^10*d^2*e^3 + 5*a^4*b^9*d*e^4 - a^5*b^8*
e^5)*x^5 + 5*(a*b^12*d^5 - 5*a^2*b^11*d^4*e + 10*a^3*b^10*d^3*e^2 - 10*a^4*b^9*d^2*e^3 + 5*a^5*b^8*d*e^4 - a^6
*b^7*e^5)*x^4 + 10*(a^2*b^11*d^5 - 5*a^3*b^10*d^4*e + 10*a^4*b^9*d^3*e^2 - 10*a^5*b^8*d^2*e^3 + 5*a^6*b^7*d*e^
4 - a^7*b^6*e^5)*x^3 + 10*(a^3*b^10*d^5 - 5*a^4*b^9*d^4*e + 10*a^5*b^8*d^3*e^2 - 10*a^6*b^7*d^2*e^3 + 5*a^7*b^
6*d*e^4 - a^8*b^5*e^5)*x^2 + 5*(a^4*b^9*d^5 - 5*a^5*b^8*d^4*e + 10*a^6*b^7*d^3*e^2 - 10*a^7*b^6*d^2*e^3 + 5*a^
8*b^5*d*e^4 - a^9*b^4*e^5)*x), -1/1920*(15*(10*B*a^5*b*d*e^4 - (3*B*a^6 + 7*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (
3*B*a*b^5 + 7*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (3*B*a^2*b^4 + 7*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*
e^4 - (3*B*a^3*b^3 + 7*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (3*B*a^4*b^2 + 7*A*a^3*b^3)*e^5)*x^2 + 5
*(10*B*a^4*b^2*d*e^4 - (3*B*a^5*b + 7*A*a^4*b^2)*e^5)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt
(e*x + d)/(b*e*x + b*d)) + (96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(28*B*a^2*b^5 + 117*A*a*b^6)*d^4*e + 4*(197*B*a^3*
b^4 + 898*A*a^2*b^5)*d^3*e^2 - 2*(278*B*a^4*b^3 + 1657*A*a^3*b^4)*d^2*e^3 + 5*(33*B*a^5*b^2 + 263*A*a^4*b^3)*d
*e^4 - 15*(3*B*a^6*b + 7*A*a^5*b^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (13*B*a*b^6 + 7*A*b^7)*d*e^4 + (3*B*a^2*b^5 +
7*A*a*b^6)*e^5)*x^4 - 10*(10*B*b^7*d^3*e^2 - (83*B*a*b^6 + 7*A*b^7)*d^2*e^3 + 2*(47*B*a^2*b^5 + 28*A*a*b^6)*d
*e^4 - 7*(3*B*a^3*b^4 + 7*A*a^2*b^5)*e^5)*x^3 + 2*(40*B*b^7*d^4*e - 2*(141*B*a*b^6 + 14*A*b^7)*d^3*e^2 + 3*(31
7*B*a^2*b^5 + 63*A*a*b^6)*d^2*e^3 - (901*B*a^3*b^4 + 609*A*a^2*b^5)*d*e^4 + 64*(3*B*a^4*b^3 + 7*A*a^3*b^4)*e^5
)*x^2 + 2*(240*B*b^7*d^5 - 8*(143*B*a*b^6 - 3*A*b^7)*d^4*e + 2*(1041*B*a^2*b^5 - 76*A*a*b^6)*d^3*e^2 - 3*(529*
B*a^3*b^4 - 139*A*a^2*b^5)*d^2*e^3 + 2*(257*B*a^4*b^3 - 342*A*a^3*b^4)*d*e^4 - 5*(21*B*a^5*b^2 - 79*A*a^4*b^3)
*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^5 - 5*a^6*b^7*d^4*e + 10*a^7*b^6*d^3*e^2 - 10*a^8*b^5*d^2*e^3 + 5*a^9*b^4*d
*e^4 - a^10*b^3*e^5 + (b^13*d^5 - 5*a*b^12*d^4*e + 10*a^2*b^11*d^3*e^2 - 10*a^3*b^10*d^2*e^3 + 5*a^4*b^9*d*e^4
- a^5*b^8*e^5)*x^5 + 5*(a*b^12*d^5 - 5*a^2*b^11*d^4*e + 10*a^3*b^10*d^3*e^2 - 10*a^4*b^9*d^2*e^3 + 5*a^5*b^8*
d*e^4 - a^6*b^7*e^5)*x^4 + 10*(a^2*b^11*d^5 - 5*a^3*b^10*d^4*e + 10*a^4*b^9*d^3*e^2 - 10*a^5*b^8*d^2*e^3 + 5*a
^6*b^7*d*e^4 - a^7*b^6*e^5)*x^3 + 10*(a^3*b^10*d^5 - 5*a^4*b^9*d^4*e + 10*a^5*b^8*d^3*e^2 - 10*a^6*b^7*d^2*e^3
+ 5*a^7*b^6*d*e^4 - a^8*b^5*e^5)*x^2 + 5*(a^4*b^9*d^5 - 5*a^5*b^8*d^4*e + 10*a^6*b^7*d^3*e^2 - 10*a^7*b^6*d^2
*e^3 + 5*a^8*b^5*d*e^4 - a^9*b^4*e^5)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)*(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (281) = 562\).

Time = 0.29 (sec) , antiderivative size = 859, normalized size of antiderivative = 2.74 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {{\left (10 \, B b d e^{4} - 3 \, B a e^{5} - 7 \, A b e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \sqrt {-b^{2} d + a b e}} - \frac {150 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 700 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 1280 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 580 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} - 150 \, \sqrt {e x + d} B b^{5} d^{5} e^{4} - 45 \, {\left (e x + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 105 \, {\left (e x + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 910 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 490 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 2944 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 1530 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 790 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} + 645 \, \sqrt {e x + d} B a b^{4} d^{4} e^{5} + 105 \, \sqrt {e x + d} A b^{5} d^{4} e^{5} - 210 \, {\left (e x + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 490 \, {\left (e x + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 2048 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 1792 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 1110 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 2370 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} - 1080 \, \sqrt {e x + d} B a^{2} b^{3} d^{3} e^{6} - 420 \, \sqrt {e x + d} A a b^{4} d^{3} e^{6} - 384 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 50 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 2370 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} + 870 \, \sqrt {e x + d} B a^{3} b^{2} d^{2} e^{7} + 630 \, \sqrt {e x + d} A a^{2} b^{3} d^{2} e^{7} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 790 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 330 \, \sqrt {e x + d} B a^{4} b d e^{8} - 420 \, \sqrt {e x + d} A a^{3} b^{2} d e^{8} + 45 \, \sqrt {e x + d} B a^{5} e^{9} + 105 \, \sqrt {e x + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-1/128*(10*B*b*d*e^4 - 3*B*a*e^5 - 7*A*b*e^5)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^4 - 4*a*b^5
*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*sqrt(-b^2*d + a*b*e)) - 1/1920*(150*(e*x + d)^(9/2
)*B*b^5*d*e^4 - 700*(e*x + d)^(7/2)*B*b^5*d^2*e^4 + 1280*(e*x + d)^(5/2)*B*b^5*d^3*e^4 - 580*(e*x + d)^(3/2)*B
*b^5*d^4*e^4 - 150*sqrt(e*x + d)*B*b^5*d^5*e^4 - 45*(e*x + d)^(9/2)*B*a*b^4*e^5 - 105*(e*x + d)^(9/2)*A*b^5*e^
5 + 910*(e*x + d)^(7/2)*B*a*b^4*d*e^5 + 490*(e*x + d)^(7/2)*A*b^5*d*e^5 - 2944*(e*x + d)^(5/2)*B*a*b^4*d^2*e^5
- 896*(e*x + d)^(5/2)*A*b^5*d^2*e^5 + 1530*(e*x + d)^(3/2)*B*a*b^4*d^3*e^5 + 790*(e*x + d)^(3/2)*A*b^5*d^3*e^
5 + 645*sqrt(e*x + d)*B*a*b^4*d^4*e^5 + 105*sqrt(e*x + d)*A*b^5*d^4*e^5 - 210*(e*x + d)^(7/2)*B*a^2*b^3*e^6 -
490*(e*x + d)^(7/2)*A*a*b^4*e^6 + 2048*(e*x + d)^(5/2)*B*a^2*b^3*d*e^6 + 1792*(e*x + d)^(5/2)*A*a*b^4*d*e^6 -
1110*(e*x + d)^(3/2)*B*a^2*b^3*d^2*e^6 - 2370*(e*x + d)^(3/2)*A*a*b^4*d^2*e^6 - 1080*sqrt(e*x + d)*B*a^2*b^3*d
^3*e^6 - 420*sqrt(e*x + d)*A*a*b^4*d^3*e^6 - 384*(e*x + d)^(5/2)*B*a^3*b^2*e^7 - 896*(e*x + d)^(5/2)*A*a^2*b^3
*e^7 - 50*(e*x + d)^(3/2)*B*a^3*b^2*d*e^7 + 2370*(e*x + d)^(3/2)*A*a^2*b^3*d*e^7 + 870*sqrt(e*x + d)*B*a^3*b^2
*d^2*e^7 + 630*sqrt(e*x + d)*A*a^2*b^3*d^2*e^7 + 210*(e*x + d)^(3/2)*B*a^4*b*e^8 - 790*(e*x + d)^(3/2)*A*a^3*b
^2*e^8 - 330*sqrt(e*x + d)*B*a^4*b*d*e^8 - 420*sqrt(e*x + d)*A*a^3*b^2*d*e^8 + 45*sqrt(e*x + d)*B*a^5*e^9 + 10
5*sqrt(e*x + d)*A*a^4*b*e^9)/((b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*((
e*x + d)*b - b*d + a*e)^5)

Mupad [B] (verification not implemented)

Time = 11.24 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.80 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {7\,{\left (d+e\,x\right )}^{7/2}\,\left (7\,A\,b^2\,e^5-10\,B\,d\,b^2\,e^4+3\,B\,a\,b\,e^5\right )}{192\,{\left (a\,e-b\,d\right )}^3}-\frac {\sqrt {d+e\,x}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,b^2}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {b^2\,{\left (d+e\,x\right )}^{9/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^4}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (21\,B\,a\,e^5-79\,A\,b\,e^5+58\,B\,b\,d\,e^4\right )}{192\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{9/2}} \]

[In]

int(((A + B*x)*(d + e*x)^(1/2))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((7*(d + e*x)^(7/2)*(7*A*b^2*e^5 + 3*B*a*b*e^5 - 10*B*b^2*d*e^4))/(192*(a*e - b*d)^3) - ((d + e*x)^(1/2)*(7*A*
b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(128*b^2) + ((d + e*x)^(5/2)*(7*A*b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(15*(a
*e - b*d)^2) + (b^2*(d + e*x)^(9/2)*(7*A*b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(128*(a*e - b*d)^4) - ((d + e*x)^(
3/2)*(21*B*a*e^5 - 79*A*b*e^5 + 58*B*b*d*e^4))/(192*b*(a*e - b*d)))/((d + e*x)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a
^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) - (d + e*x)^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*
e^2 - 30*a*b^4*d^2*e) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*
(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4
*b*d*e^4) + (e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(7*A*b*e + 3*B*a*e - 10*B*b*d))/((a*e - b*d)^(1/2)*(7*A*b*e
^5 + 3*B*a*e^5 - 10*B*b*d*e^4)))*(7*A*b*e + 3*B*a*e - 10*B*b*d))/(128*b^(5/2)*(a*e - b*d)^(9/2))